UDC 517.948.32:517.544

MATHEMATICS

V. D. FROLOV

ON THE THEORY OF SINGULAR INTEGRAL EQUATIONS WITH MEASURABLE COEFFICIENTS

(Presented by Academician N. I. Muskhelishvili on 20 V 1969)

Let a simple closed oriented Lyapunov contour \(\Gamma\) divide the complex plane into an exterior domain \(D^-\) and an interior domain \(D^+\), and let the point \(z=0\) belong to \(D^+\). By \(\Lambda\) we denote the set of functions piecewise continuous on \(\Gamma\), and by \(\mathscr L\) the closure of this set in the sense of uniform convergence. Following \(\left({}^{1}\right)\), we shall call functions in \(\mathscr L\) lineatchatye.

In the present note we investigate the problem of perturbations of a singular operator with bounded measurable coefficients and establish a general theorem on the solvability of singular integral equations in the space \(L_p\) with coefficients from \(\mathscr L\).

Consider the singular integral equation in the space \(L_p(\Gamma)\)
\[ d(t)(S\varphi)(t)+c(t)\varphi(t)=g(t), \tag{1} \]
where \(c(t)\) and \(d(t)\) are measurable and essentially bounded functions on \(\Gamma\), and
\[ (S\varphi)(t)=\frac{1}{\pi i}\int_\Gamma \frac{\varphi(\tau)}{\tau-t}\,d\tau \quad (t\in\Gamma). \]

With the aid of the projectors \(P=(I+S)/2\) and \(Q=(I-S)/2\), the integral equation (1) is rewritten in the following form:
\[ aP+bQ=g, \]
where \(a(t)=c(t)+d(t)\), \(b(t)=c(t)-d(t)\).

Theorem 1. Let \(a(t)\) be a bounded measurable function and let the operator \(A=aP+Q\) be a \(\Phi\)-operator in the space \(L_p(\Gamma)\). Then there exists a number \(r\) \((1<r<\infty)\) such that, for any function \(c(t)\) of the class \(A[r]\), the operator \(B=acP+Q\) is a \(\Phi\)-operator, and
\[ \dim\ker B=\max(0,\operatorname{ind}A-\operatorname{ind}c),\qquad \dim\operatorname{coker}B=\max(0,\operatorname{ind}c-\operatorname{ind}A). \]

Proof. For the \(\Phi\)-operator \(A\) there exists a number \(\eta>0\) \(\left({}^{3}\right)\) such that any operator \(A_0\) for which \(\|A_0-A\|<\eta\) is a \(\Phi\)-operator and \(\operatorname{ind}A_0=\operatorname{ind}A\). Choose a number \(r(>1)\) from the condition
\[ \max[r,r']>\pi\sup |a|\,\|P\|/\eta, \]
where \(1/r+1/r'=1\).

Let now \(c(t)\) be any function of the class \(A[r]\); then \(\left({}^{2}\right)\)
\[ c(t)=c_1(t)c_2(t), \]
where \(c_1(t)\) is a Hölder function and \(\operatorname{ind}c_1=\operatorname{ind}c=\chi\), while \(c_2(t)\) is a function of the class \(A[r]\) and
\[ |\arg c_2|<\pi/\max[r,r']-\varepsilon. \]
Since for the function
\[ m(t)=c_2(t)/|c_2(t)|-1 \]
we have
\[ |m(t)|<\sup_{t\in\Gamma}|\arg c_2(t)|<\eta/\sup |a|\,\|P\|, \]
it follows that
\[ K=a(1+m)P+Q \]
is a \(\Phi\)-operator and \(\operatorname{ind}K=\operatorname{ind}A\).

Let \(\chi\ge 0\). Using the factorization of the functions \(c_1=f_+t^\chi f_-\) and \(|c_2|=g_+g_-\), where the functions \(f_+f_-\), \(g_+\), \(g_-\) and their inverses are bounded on the contour \(\Gamma\) (see \(\left({}^{2}\right)\)), the operator \(B\) can be represented in the form
\[ B=X\bigl(a(1+m)P+Q\bigr)Y(t^\chi P+Q), \]
where
\[ X=g_-f_-I \]
and
\[ Y=g_+f_+P+g_-^{-1}f_-^{-1}Q \]
are invertible

* For the definition of the class \(A[r]\) and the index of a function \(c(t)\in A[r]\), see \(\left({}^{2}\right)\).

operators. Taking into account that the operator \(t^\chi P+Q\) is a \(\Phi\)-operator and
\(\operatorname{ind}(t^\chi P+Q)=-\chi\), we obtain that \(B\) is a \(\Phi\)-operator,
\(\operatorname{ind} B=\operatorname{ind} A-\chi=\operatorname{ind} A-\operatorname{ind} c\), and by virtue of \((^4,^5)\) it is left invertible.

For \(\chi<0\) the equality
\[ B(t^{-\chi}P+Q)=X(a(1+m)P+Q)Y \]
holds, whence it follows that the operator \(B\) is right invertible and
\(\operatorname{ind} B=\operatorname{ind} A-\chi\). The theorem is proved.

Corollary. If \(A=aP+Q\) is a \(\Phi\)-operator, then there exists a number \(l(a,p)\) \((>0)\) such that, for every measurable function \(c(t)\) satisfying the conditions
\(0<N\le |c(t)|\le M<\infty\) and
\[ \sup_{t\in\Gamma}|\arg c(t)|<l(a,p), \]
the operator \(B=acP+Q\) will be a \(\Phi\)-operator and \(\operatorname{ind} B=\operatorname{ind} A\).

For the case of a piecewise continuous function \(a(t)\), one can indicate the dependence of the quantity \(l(a,p)\) on the function \(a(t)\) and the space \(L_p\). First let us note some properties of functions from \(\Lambda\) and \(\mathscr L\).

Following \((^6)\), to each function \(a(t)\in\mathscr L\) we associate, in the natural way, the curve with orientation
\[ V_p(a)(\mu,t)=a(t-0)M(p,\mu)+a(t+0)(1-M(p,\mu)) \]
\[ (0\le \mu\le 1,\ t\in\Gamma), \]
where
\[ M(p,\mu)=\exp\{(1-\mu)\theta\}\sin\mu\theta/\sin\theta,\quad \theta=\pi(1-2/p), \]
if \(p\ne2\), and \(V_2(a)=a(t-0)\mu+a(t+0)(1-\mu)\). A function \(a(t)\) from \(\mathscr L\) is called \(p\)-nonsingular if
\[ \inf_{t,\mu}|V_p(a)|>0. \]

Let \(a(t)\) be a \(p\)-nonsingular function from \(\mathscr L\), \(t_0\) an arbitrary point of continuity, and \(t_1,t_2,\ldots\) its discontinuity points; then the ratio \(a(t_k-0)/a(t_k+0)\) can be represented in the form
\[ |a(t_k-0)/a(t_k+0)|\exp\{ih_k(a)\}\quad(k=0,1,2,\ldots), \]
where
\[ -2\pi/q<h_k(a)<2\pi/p,\qquad 1/p+1/q=1. \]
To each such function one can assign two numbers \(h_+(a)\) and \(h_-(a)\), defining them by the equalities
\[ h_+(a)=\max_k h_k(a),\qquad h_-(a)=\min_k h_k(a). \]
Obviously,
\[ -2\pi/q<h_-(a)\le h_k(a)\le h_+(a)<2\pi/p. \tag{2} \]

For \(p\)-nonsingular functions from \(\mathscr L\) the index is introduced in the following way. Let a sequence of functions \(a_n(t)\) from \(\Lambda\) converge uniformly to \(a(t)\). Then, starting from some \(n\), the functions \(a_n(t)\) are \(p\)-nonsingular and their \(p\)-index is equal to one and the same number; this number is taken to be the \(p\)-index of the function \(a(t)\) from \(\mathscr L\).

Theorem 2. For a \(p\)-nonsingular function \(a(t)\) from \(\Lambda\), as the number \(l(a,p)\) one may take
\[ \min[2\pi-ph_+(a),\ 2\pi+qh_-(a)]/2\max[p,q]\qquad (1/p+1/q=1). \]
Moreover, if \(\chi(=\operatorname{ind}_p a)=0\), the operator \(A=acP+Q\) is invertible in the space \(L_p(\Gamma)\); if \(\chi>0\), the operator \(A\) is left invertible and \(\dim\operatorname{coker} A=\chi\); if \(\chi<0\), the operator \(A\) is right invertible and \(\dim\ker A=|\chi|\).

Remark 1. The number \(l(a,p)\) indicated in the theorem is, in a certain sense, sharp. Indeed, take on the unit circle \(\Gamma_0\) the piecewise continuous functions
\[ c=a=\exp\{i\theta/4\}\qquad (-\pi<\theta\le\pi); \]
then
\[ \max_{\in\Gamma_0}|\arg c|=l(a,p), \]
and the operator \(caP+Q\) in the space \(L_2(\Gamma_0)\) is not invertible from either side (see \((^6)\)).

Proof of Theorem 2. From the discontinuity points \(t_k\) of the function \(a(t)\) and the numbers \(h_k(a)\) we construct the function
\[ \psi(t)=\prod_{k=1}^{m} t^{\gamma_k},\qquad \text{where }\gamma_k=\frac{h_k(a)}{2\pi}+\frac{1}{2\pi i}\ln\left|\frac{a(t_k-0)}{a(t_k+0)}\right|, \]
and the function \(t^{\gamma_k}\) has a discontinuity only at the point \(t_k\) \((k=1,2,\ldots,m)\). Then the function \(b(t)=a(t)\psi^{-1}(t)\) is continuous on the contour \(\Gamma\), and
\(\operatorname{ind} b=\operatorname{ind}_p a\). Using the condition of the theorem, choose a number \(\varepsilon>0\) so that
\[ |\arg c|< (\min[2\pi-ph_+(Q),\ 2\pi+qh_-(Q)]-\varepsilon)/2\max[p,q]. \]
Let \(r(t)\) be such—

a rational function such that \(b(t)=r(t)(1+m(t))\), \(\operatorname{ind} r(t)=\operatorname{ind} b(t)\),
\(|\arg(1+m)|<\varepsilon/8\max[p,q]\). The functions \(r,\psi\), and \(c_1=c(1+m)\) admit the factorization \((^2,^7)\)

\[ r=r_{+}t^{\chi}r_{-},\qquad \psi=\psi_{+}\psi_{-},\qquad c(1+m)=c_{+}c_{-}. \]

The factorization factors for the functions \(\psi\) and \(c_1\) can be represented in the form

\[ \psi_{\pm}=F_{\pm}\rho_1^{\pm1},\quad \text{where }\sup_t |F_{\pm}^{\pm1}(t)|<\infty,\quad \rho_1(t)=\prod_{k=1}^{m}|t-t_k|^{\operatorname{Re}\gamma_k}, \]

\[ c_{\pm}=u_{\pm}\rho_2^{\pm1},\quad \text{where }\sup_t |u_{\pm}^{\pm1}(t)|<\infty,\quad \rho_2(t)=\exp\left\{\frac{1}{2\pi}\int_{\Gamma}\frac{\arg c_1(\tau)}{\tau-t}\,d\tau\right\}. \]

Consider the operator \(Z=(\rho_1\rho_2)^{-1}S\rho_1\rho_2\); its boundedness in the space \(L_p(\Gamma)\) is proved by applying Stein’s theorem \((^8)\). Indeed, take
\(\gamma=2\pi/(\min[2\pi-ph_{+}(a),\ 2\pi+qh_{-}(a)]-\varepsilon/2)(>1)\); then
\(|\gamma\arg c_1(t)|<\pi/\max[p,q]-\varepsilon_1\) \((\varepsilon_1>0)\), and by I. B. Simonenko’s theorem on weights \((^2)\)

\[ \|(Sf)\rho_2^{-\gamma}\|_{L_p}\leq M_1\|f\rho_2^{-\gamma}\|_{L_p}. \tag{3} \]

Moreover, since the function \(a(t)\) is \(p\)-nonsingular, for the chosen \(\gamma\) the relations

\[ -1/q<\operatorname{Re}\gamma_k\cdot \gamma/(\gamma-1)<1/p\quad (k=1,2,\ldots,m) \]

follow from (2).

Then from B. V. Khvedelidze’s theorem on the boundedness of singular operators in \(L_p\) spaces with a weight \(((^7),\text{ p. }24)\) it follows that

\[ \|(Sf)\rho_1^{-\gamma/(\gamma-1)}\|_{L_p}\leq M_2\|f\rho_1^{-\gamma/(\gamma-1)}\|_{L_p}. \tag{4} \]

Since for \(t=1/\gamma\)
\[ \rho_2^{-\gamma t}\rho_1^{-(1-t)\gamma/(\gamma-1)}=(\rho_2\rho_1)^{-1}, \]
from (3) and (4), by virtue of Stein’s interpolation theory, the boundedness of the operator \(Z\) follows.

Let \(\operatorname{ind}_p a=\chi=0\); then the operator \(A\) can be represented in the form

\[ A=acP+Q=r_{-}\psi_{-}c_{-}(r_{+}\psi_{+}c_{+}P+r_{-}^{-1}\psi_{-}^{-1}c_{-}^{-1}Q). \]

Consider the operator

\[ B=(r_{+}^{-1}\psi_{+}^{-1}c_{+}^{-1}P+r_{-}\psi_{-}c_{-}Q)r_{-}^{-1}\psi_{-}^{-1}c_{-}^{-1}. \]

Its boundedness follows easily from the boundedness of the operator \(Z\). It is not difficult to verify that the operator \(B\) is inverse to \(A\). Thus, for \(\chi=0\) the operator \(acP+Q\) is invertible in \(L_p(\Gamma)\).

Let \(\chi>0\); then \(\operatorname{ind}(at^{-\chi})=0\), and from what was proved above it follows that the operator \(t^{-\chi}acP+Q\) is invertible in \(L_p(\Gamma)\). Since the operator \(t^{\chi}P+Q\) is left invertible, it follows from the equality
\[ (act^{-\chi}P+Q)(t^{\chi}P+Q)=acP+Q \]
that the operator \(A\) is left invertible and \(\dim\operatorname{coker}A=\chi\).

If \(\chi<0\), then
\[ act^{-\chi}P+Q=(acP+Q)(t^{-\chi}P+Q), \]
and, consequently, the operator \(A\) is right invertible and \(\dim\ker A=|\chi|\).

Remark 2. The proof given above extends without difficulty to the case of a \(p\)-nonsingular function \(a(t)\) from \(\mathscr L\).

Theorem 3. Let \(a(t), b(t)\in\mathscr L\). In order that the operator
\[ \hat A=aP+bQ \]
be a \(\Phi_{+}(\Phi_{-})\)-operator in the space \(L_p(\Gamma)\), it is necessary and sufficient that: 1) \(\operatorname*{ess\,inf}_{t\in\Gamma}|b(t)|>0\); 2) the function \(a(t)/b(t)\) be \(p\)-nonsingular.

If these two conditions are fulfilled and \(\chi=\operatorname{ind}_p(a/b)\), then for \(\chi=0\) the operator \(\hat A\) is invertible in the space \(L_p(\Gamma)\); for \(\chi>0\) the operator \(\hat A\) is left invertible and \(\chi=\dim\operatorname{coker}\hat A\); for \(\chi<0\) the operator \(\hat A\) is right invertible and \(|\chi|=\dim\ker\hat A\).

Proof. The sufficiency of the conditions of the theorem follows from Remark 2. Necessity is established as follows. Let \(\hat A=aP+bQ\) be a \(\Phi_+(\Phi_-)\)-operator and
\[ \operatorname*{ess\,inf}_{t\in\Gamma}|b(t)|=0. \]
Since the set of \(\Phi_+(\Phi_-)\)-operators is open, one can choose functions \(\tilde a\) and \(\tilde b\) from \(\Lambda\) such that
\[ \operatorname*{ess\,inf}_{t\in\Gamma}|\tilde b(t)|=0 \]
and \(\tilde aP+\tilde bQ\) is a \(\Phi_+(\Phi_-)\)-operator, which leads to a contradiction with (6). Thus
\[ \operatorname*{ess\,inf}_{t\in\Gamma}|b(t)|>0. \]
Now the problem reduces to the study of the operator
\[ A_0=(a/b)P+Q. \]
Let \(A_0\) be a \(\Phi_+(\Phi_-)\)-operator and let \(a/b\) not be a \(p\)-nonsingular function. Then there exists a function \(g(t)\in\Lambda\) such that
\[ \inf_{t\in\Gamma,\ 0\le \mu \le 1}|V_p(g)(\mu,t)|=0 \]
and the operator \(gP+Q\) is a \(\Phi_+(\Phi_-)\)-operator. This leads to a contradiction. The theorem is proved.

Let us note that the sufficiency of the conditions of Theorem 3 for the case when
\[ \sum |h_k(a)|<\infty \]
is established in \((^9)\).

Institute of Mathematics with Computing Center
Academy of Sciences of the MSSR
Kishinev

Received
5 V 1969

CITED LITERATURE

1. N. Bourbaki, Functions of a Real Variable, “Nauka,” 1965.
2. I. B. Simonenko, Izv. AN SSSR, 28, No. 2 (1964).
3. I. Ts. Gokhberg, M. G. Krein, UMN, 12, issue 2 (1957).
4. I. Ts. Gokhberg, DAN, 122, No. 3 (1958).
5. I. B. Simonenko, Izv. AN SSSR, 32, No. 5 (1968).
6. I. Ts. Gokhberg, N. Ya. Krupnik, Studia Math., 31, 347 (1968).
7. B. V. Khvedelidze, Tr. Tbilissk. matem. inst., 23, 3 (1957).
8. E. M. Stein, Trans. Am. Math. Soc., 83, No. 2 (1956).
9. V. Yu. Shelepov, DAN, 181, No. 3 (1968).